exo7 4357

Soient \(J(x) = \int_{t=0}^{\pi/2}\frac{d t}{\sqrt{\sin^2t + x^2\cos^2t}}\) et \(K(x) = \int_{t=0}^{\pi/2}\frac{\cos t\,d t}{\sqrt{\sin^2t + x^2\cos^2t}}\).

Calculer \(\lim_{x\to0^+}(J(x)-K(x))\) et montrer que \(J(x) = -\ln x + 2\ln2 + o_{x\to0^+}(1)\).